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Cosmological Nucleosynthesis

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Cosmological Nucleosynthesis Cosmology Cosmological Nucleosynthesis v In the discussion of the thermal history of the early universe form temperatures 1011 ◦K down to about 106 ◦K we were neglecting the tiny contribution of relict nucleons which were not completely annihilated in particle-antiparticle annihilation. v Nucleons, the building blocks of atomic nuclei, were formed abundantly in the QCD phase transition out of the quark-gluon plasma at temperatures T ∼ 150 MeV corresponding to a temperature T ∼ 2 × 1012 ◦K and cosmic time t ∼ 20µs. v The QCD phase transition sets in after the annihilation of the heavier quarks (t; b; c) during the annihilation of s quarks. During the QCD phase transition color neutral SU(3)c triplet states form, the baryons, as well as anti-triplets representing the antibaryons together with quark–antiquark singlets, the mesons. The lightest hadrons are the nucleons (N), proton (p) and neutron (n), and their antiparticles (spin 1/2 fermions), which when formed are non-relativistic (essentially at rest) and the lightest mesons, the pions π± and π0 (spin 0 bosons). c 2009, F. Jegerlehner nx Lect. 9 xo 138 Cosmology v Baryon–antibaryon annihilation immediately sets in during their formation. Here the mystery of the baryon asymmetry plays the key role for the future evolution, namely, the annihilation turned out to be incomplete with a relict −10 −9 η = nN(t0)=nγ(t0) ∼ 10 − 10 of surviving baryons. The only particles left in large numbers at this time were pions, muons, electrons, neutrinos and photons. After baryon–antibaryon annihilation, for T < 10 MeV, the total number of baryons nB = nN − nN¯ = np − np¯ + nn − nn¯, in practice nB = np + nn since now nN¯ nN, remains conserved due to baryon number conservation. Baryons are free protons, free neutrons or protons and neutrons bound in nuclei. While nuclear strong interaction forces bind nucleons in nuclei weak interactions convert neutron and protons into each other. r As we will see, about one quarter of all nucleons form nuclei (A > 1), with about an equal amount of neutrons and protons, while approximately three quarters remain free protons. c 2009, F. Jegerlehner nx Lect. 9 xo 139 Cosmology + − r Later, when T < 10 keV, after e e –annihilation, we also will have ne+ ne− and ne− = np, where np represents all protons, the free ones as well as the ones bound in nuclei (presumed charge neutrality of the universe). r Radiation and matter (electron, protons and nuclei) remain in thermal equilibrium as long as there are lots of free electrons. After e+e−–annihilation, atoms in particular hydrogen may form e− + p H + γ and essentially, when Tγ < 13:6 eV, the ionization energy of hydrogen, electrons will have been trapped by protons and nuclei ions (recombination). The universe becomes transparent and photons decouple from matter. More details on that later. We are now discussing the evolution of the relict densities of baryons and the formation of light nuclei. Relevant basic processes: c 2009, F. Jegerlehner nx Lect. 9 xo 140 Cosmology n + ν p + e− ; n + e+ p + ν¯ ; n p + e− + ν¯ v ν ≡ νe, other neutrinos inactive here. v kB T mN ! nucleons at rest (non-relativistic) v Q–value: Q = mn − mp ' 1:293 MeV − Q = Ee − Eν n + ν p + e + Q = Eν − Ee n + e p + ν¯ − Q = Eν + Ee n p + e + ν¯ Calculation of rate: r cross-section: weak 2–body reaction n + e ! p + ν¯ 2 2 2 2π2~3 AE2 G (1+3 g ) cos θC σ = ν ; A = F A ve 2π3 ~ c 2009, F. Jegerlehner nx Lect. 9 xo 141 Cosmology −5 −2 GF = 1:16637(1) × 10 GeV Fermi constant gA = 1:257 nucleon axial vector coupling of β-decay cos θC = 0:9745(6) , ΘC Cabibbo mixing angle ve the electron velocity Electrons and neutrinos are relativistic ! have to take into account Fermi distribution (Pauli-Principle): r number density of electrons (with momentum between pe and pe + dpe) 2 4 π pe dpe exp (E =k T) + 1 −1 (2 π ~)3 e B c 2009, F. Jegerlehner nx Lect. 9 xo 142 Cosmology r fraction of unfilled antineutrino levels: −1 −1 1 − exp (Eν=kBTν) + 1 = exp (−Eν=kBTν) + 1 such that the total rate reads 1 + R 2 2 −1 −1 λ(n + e ! p + ν¯) = A Eν pe dpe exp (Ee=kBT) + 1 × exp (−Eν=kBTν) + 1 0 changing variable to q ≡ −Eν = −Q − Ee yields an integral over q + v from q = −∞ to q = −Q − me for n + e ! p + ν¯ − v from q = −Q + me to q = 0 for n ! p + e + ν¯ v from q = 0 to q = +1 for n + ν ! p + e− while the integrand is unchanged. The total rate of disappearance of neutrons then is given by c 2009, F. Jegerlehner nx Lect. 9 xo 143 Cosmology +1 r R m2 (Q+q)2 q2 dq λ(n ! p) = = 1 − e (Q+q)2 1+eq=kBTν 1+e−(Q+q)=kBT −∞ with a gap from q = −Q − me to q = −Q + me in the integration range (unphysical region). Similarly, one calculates the total rate of production of neutrons to be +1 r R m2 (Q+q)2 q2 dq λ(p ! n) = = 1 − e (Q+q)2 1+e−q=kBTν 1+e(Q+q)=kBT −∞ again, with a gap from q = −Q − me to q = −Q + me in the integration range. This allows us to calculate the fraction of neutrons among the nucleons, by solving the differential equation dXn dt = −λ(n ! p) Xn + λ(p ! n) (1 − Xn) c 2009, F. Jegerlehner nx Lect. 9 xo 144 Cosmology with appropriate initial conditions. Initial Condition One can check (Exercise) that for fixed time–independent temperature T = Tν the two rates λ(p ! n) and λ(n ! p) have a ratio λ(p!n) λ(n!p) = exp (−Q=kBT) for T = Tν which would imply a time–independent solution c 2009, F. Jegerlehner nx Lect. 9 xo 145 Cosmology Xn = Xn = exp (−Q=k T) ; Xp 1−Xn B the non-relativistic equilibrium distribution. Thus what keeps this ratio away from equilibrium is the inequality of T and Tν and the fact that they depend on time. When baryons are produced during the QCD–phase transition we have kBT Q and we can calculate the integrals by setting T = Tν and Q; me = 0, such that +1 Z q4 dq λ(n ! p) = λ(p ! n) = A 1 + eq=kBT 1 + e−q=kBT −∞ 7 T 5 = π4 A (k T)5 = 0:400 sec−1 15 B 1010 ◦K which is > 1 for T > 1:22 × 1010 ◦K . c 2009, F. Jegerlehner nx Lect. 9 xo 146 Cosmology Thermal equilibrium means dXn λ(p!n) 1 = 0 , Xn = , Xn = dt λ(p!n)+λ(n!p) (1+exp(Q=kBT)) Crucially important is that the equilibrium value is established at high enough temperatures when the electron chemical potential is zero (still large numbers of electrons and positrons) such that no assumption about the initial proton/neutron ratio is necessary! This is expected to be satisfied for T > 3 × 1010 ◦K and we may solve our exact equation for Xn with this boundary condition. The solution can be obtained numerically; see the Table c 2009, F. Jegerlehner nx Lect. 9 xo 147 Cosmology ◦ T( K) t(sec) Xn 1 × 1012 0:0001 0:4962 3 × 1011 0:0011 0:4875 1 × 1011 0:0099 0:4626 3 × 1010 0:1106 0:3798 1 × 1010 1:008 0:2386 3 × 109 12:67 0:1654 1:3 × 109 91:09 0:1458 1:2 × 109 110:2 0:1425 1:1 × 109 135:1 0:1385 1 × 109 168:1 0:1333 9 × 108 2:12:7 0:1268 8 × 108 274:3 0:1182 7 × 108 362:6 0:1070 6 × 108 496:3 0:0919 3 × 108 1980 0:0172 1 × 108 17780 3:07 × 10−10 c 2009, F. Jegerlehner nx Lect. 9 xo 148 Cosmology Given the rate λ, the maximum number of collisions can be found by multiplying the rate with the Hubble time H = 1=2t. Earlier, we calculated s !2 3 1 1010 ◦K t = + const = 0:994 sec + const 16πGag∗ T 2 T 3 such that, with g∗ = 43=4, λ/H ' 0:8 × T which is > 1 for 1010 ◦K T > 1:1 × 1010 ◦K . Because of the T 3 behavior the rate drops fast and equilibrium is lost quickly between 3 × 1010 ◦K and 1010 ◦K . Little later all but neutron β–decay plays a role in proton– neutron transmutation, also due to the fact that electron and positron densities rapidly decrease because of e+e−–annihilation. Then the decrease of Xn just follows neutron decay. Exercise: Show that Xn ! 0:1609 exp (−t/τn) with τn = 885:7 sec is the neutron lifetime. The above would be the story if nucleons would not get bound into nuclei where also neutrons get stable. c 2009, F. Jegerlehner nx Lect. 9 xo 149 Cosmology Nuclei if in Equilibrium We are interested here in what happens after temperatures have dropped way below the proton mass T mp such that all nucleons and nuclei are non-relativistic. In thermal equilibrium we then have (see Exercise at end of previous lecture) m T 3=2 µi−mi i T ni = gi 2π e for the number densities of proton (i = p), neutron (i = n) or a nuclei of type i = (Ai − Zi) n + Zi p. If nuclear reaction rates for the formation of a nucleus (Ai − Zi) n + Zi p i from protons and neutrons would be sufficiently high to maintain chemical equilibrium, we would have µi = (Ai − Zi) µn + Zi µp c 2009, F.
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